.• u-i

IC/66/81

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

ON FACTORIZATIONOF EINSTEIN'S FORMALISM INTO A PAIR

OF QUATERNION FIELD EQUATIONS

M. SACHS

1966PIAZZA OBERDAN

TRIESTE

IC/66/S1

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON FACTORIZATION OF EINSTEIN'S FORMALISM

INTO A PAIR OF QUATERNION FIELD EQUATIONS *"*

M. SACHS**

TRIESTE

July 1966

' To be submitted to Nuovo Cirnento* The research reported in this paper has been sponsored by the Air Force Cambridge Research Laboratories, Office of

Aerospace Research, under Contract AF 19(628)-2816** On leave of absence from Dept. of Physics, Boston University, Boston, Mass., USA

Permanent address after September, 1966: Dept. of Physics, State University of New York, Buffalo, N.Y., USA

ABSTRACT

It is proposed that a full exploitation of the principle of general

relativity in the construction of the metrical field equations implies that

the fundamental variables should be quaternion fields rather than the

metric tensor field of the conventional formulation. Thus, the tensor

property of Einstein's formalism is replaced here by a formalism that

transforms as a quaternion - a vector field in co-ordinate space and a

second-rank spinor field of the type >? 383 >\ in spinor space. The geo-

metrical field variables of the Riemann space are derived in quaternion

form. The principle of least action ^vith the Palitini technique) is then used

to derive a pair of time-reversed quaternion field equations, from the

(quaternionic form of ) Einstein1 s Lagrangian, It is then shown how the

conventional tensor form of the Einstein formalism is recovered from a

particular combination of the derived time-reversed quaternion equations.

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ON FACTORIZATION OF EINSTEIN'S FORMALISM INTO A PAIR OF

QUATERNION FIELD EQUATIONS

1. INTRODUCTION

Shortly after Dirac1 s discovery of the special relativistic spinor

wave equation, several investigations were initiated to study the role of

spinor variables in a generally relativistic formalism. It was found that,

indeed, the Riemannian manifold could alternatively be described in terms

of four four-vector fields at each space-time point (i. e., a tetrad field)

rather than the usual metric tensor field.

In close connection with the latter formalism one may also con-

struct a theory of general relativity in which fundamental quaternion fields2

(and their conjugates) play the role of defining the metric field. Using the

c orres pondence

(1) <j°

or equivalently

(I1) ff f q f

where cr° is the unit two-dimensional matrix and "Tr" denotes the trace

the Einstein field equations

(2) R*u - i g ^ R = KT^

can be re-expressed in a form in which the quaternion variables fy ("*•> > ^

rather than g/1' X) are the field solutions of these equations. The quater-

nion variables have the structure

where vv^ are unit four-vectors with j/M' having a positive norm and hv/t*'

having negative norms. The conjugate quaternion to 0/ is its time-

reversed field, i .e . , the quaternionic basis elements for QM are (o"k* er°)

while those of q/ are (0" )"(T ) where or * are the usual three Pauli

matrices. This definition of quaternion conjugation is equivalent to the

following relationship:

(3)

where e - (_\ J

and the asterisk denotes complex conjugation. Finally, it is readily veri-

fied that with the normalization of ^V* defined above, an invariant of the

Riemannian manifold is

(4) q^ V s q^ q A = " 4

It is important to note that the quaternion fields transform in co-

ordinate space as four-vectors and that they have additional transformation

properties in an "internal space" - both the quaternion and its conjugate

behave in spinor space as a second-rank spinor of the type V) H i *".

An implication of the property of q/UV as a bilinear product in

< and "a. , is that the Einstein field equations (in o " ^ ) might factor

into two (time-reversed) equations which separately transform as quaternion

fields. The purpose of this paper is to demonstrate such a "factorization".

2. THE VARIABLES OF THE RIEMANN SPACE IN QUATERNION

FORM

It follows from the invariance of the metric Q^^v that the

covariant derivatives of the quaternion fields must vanish. The covariant

derivative, in turn, depends on two distinct parts - one that relates to a

change in spinor space and the other to a change in co-ordinate space.

Denoting the former degrees of freedom in terms of the spinor components

of the quaternion and the latter by [ P j , we have

(5)

where the cova riant derivative of a spinor field w defines the spin-affine

connection ^ i p as follows

(6) rj. p = (9j> +n_p)»7

It is readily verified that as a consequence of the vanishing of the covariant

derivatives of Q K , "W, has the two equivalent forms

(7) Q?* \ fyq* + r^pqT) q& = - ~ q * *

where | PT i is the ordinary affine connection.

Combining Eqs. (6) and (5), the following relationship follows

Since the last term on the right-hand side of Eq. (8) refers only to the behaviour

of the quaternion as a vector in co-ordinate space, it relates to the Riemann

curvature tensor , i. e.,

0) fV];f;*" [ V ] ^ ^ = R/"tj*<i*

With the definition (6) of spin-affine connection and denoting the spin curvature

by Kc,\ > we have

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Substituting Eqs. (9) and (10) into Eq. (8) frnd dropping the spinor component

indices) the following equation results

(Ha) (KfX q^ + qAK^x) = - R^HJ* q *

where the "dagger" denotes the hermitian adjoint. It is also readily veri-

fied from the vanishing of the covariant derivatives of the conjugate quater-

nion and the relationship (3) that the time-reversed equation which accom-

panies Eq. (lla) is

(lib) Kf* V1" VK i* ' R^^qK

Finally, if we multiply Eq. (lla) on the right with cj, and Eq. (lib)

on the left with <^K , add the resulting equations and make use of the follow-

ing identity (for K fixed)

the following expression for the Riemann curvature tensor results

(12) <?° RAHfA = | [Kj,* q , ^ - qK q^ Kfx + q^ K J A qK - q,

Taking the trace of both sides of Eq. (12) the coefficients of the curvature

tensor can then be expressed in the form

(12') RAKpA = ~ Tr iKpAiq^ qK - qK q J + h. c.

where "h. c. " denotes the hermitian conjugate matrix field.

The Ricci tensor may now be obtained in the usual way by con-

tracting tWKpX with the contravariant metric tensor

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RKf = \ [KpX q* qK - qK qA Kf\

(13)

The trace of this equation then gives the components of the Ricci tensor

(13') RKf =±Tr[Kj,*{q*qk - qK qX) + h. c. ]

Finally, the scalar curvature of the Riemann space is

R = gK?RKj) =^TrtKj?A(q'1 qP - qJ5 q*) + h . c ]

If we use the fact that

the scalar curvature then reduces to the form

(14) R = | Tr [q> K ^ q* + h. c. ]

To sum up, the respective expressions in a Riemann space for the

metric tensor, curvature tensor, Ricci tensor and scalar curvature, in

terms of the spinor-quaternion formalism, are given in Eqs. (1), (12), (13)

and (14).

3. DERIVATIONS OF THE METRICAL FIELD EQUATIONS FROM

THE PRINCIPLE OF LEAST ACTION

3Einstein's well-known derivation of the field Eq. (2) follows from

a minimization of the action functional

(15) A = A 6+AH

- 6 -

where

(16) AG = f/gd* x = f R /^g d* x

and A^ is the contribution due to matter and electromagnetic fields. In

this variational procedure it is assumed that only the components of the

metric tensor q^ are the independent field variables and that all fields

are non-singular.

4There is also the alternative technique of Palitini for deriving

the field Eq. (2) from the minimization of oC which takes both the metric :

tensor Q, v and the components of the affine connection fl vf to be the in-

dependent variables. The latter technique gives two sets of field equations.

The first, in q^ , does not relate the metric tensor to the affine connect-

ion. , The second, in f \, , leads to the vanishing of the covariant de-

rivatives of the metric tensor field which, in turn, implies the mathematical

relation between P*, and derivatives of g^ . Thus the Palitini technique

gives field equations that are equivalent to Einstein's equations; yet it treats

the coefficients (P v 1 in a more general way thereby allowing the pos-

sibility of extending the theory by generalizing the affine connection.

In the derivation that follows, the Palitini technique will be

utilized to derive the quaternionic form of the metrical field equations. The

Lagrangian will be taken, in the variational calculation, as a function of the

independent variables

and their conjugates, where the following notation is used :

(17) ^Wf a (8J>+ «/) *K»

The spin curvature in Eq. (10) is then expressible in the form

(18)

If # and B are any two matrix fields, it follows that

a 9 t- Tr(AB) = Aii or -5-5 T^AB).55 A1

j J 9B

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AAV

Using the correspondence (1) between q and the quaternion variables,

the derivatives of the metric density with respect to the latter fields are as

follows

r ("4 T r

(20)

With the relationship (19) in Eq. (20) and the hermitian property of the

quaternions it follows that

Finally, with the relationship (14) in the Lagrangian

using Eq. (21) and the anti- symmetric property of the spin curvature

it follows that

(22a) f | ? = [ | (qA KTpA + KpX q>) + | R q f]* sf- g

In a similar fashion, the following accompanying time-reversed relation is

obtained:

(22b) f e = [ - \ (KTp qA + q* KfK) + I R ^ ] * / - g

Since dCG does not have any explicit dependence on derivatives

of the quaternion fields, the substitution of Eqs. (22) into the Lagrange-

Euler formalism yields the following field equations;

(23a) 7 (KpX q* + qX KJJ ) + £ R qp = K TP(K^ q + q Kjj ) + | R qf

(23b) - | (KJA qX + q Kf*) + | R q = K(rO) )

where 0 , is the (complex conjugate of the) vector field which is obtained

from a minimization (with respect to the quaternion variables) of the rest

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of the Lagrangian that depends on the matter variabLes. The right-hand

side of Eq. (23b)

is the time-reversal of the source field X in Eq. (23a).

Finally, we shall shall determine the relationship between the

spin-affine connection oi^ and the quaternion fields from the independent

variation of oT with respect to Sl^ and *>l<L\ ^ . Since X^ depends

on the affine connection through the spin curvature K e which in turn

depends only on ti^-p, and not <>>l« itself (Eq. (18), the variational procedure

applied to these fields gives the following result:

&.L - «rWith Q r and a r arbitraryj the preceding equation implies that the covariant

derivatives of the quaternion fields themselves must be zero. This result, in

turn, leads to the relationship (7) between the spin-affine connection and the

derivatives of the fundamental quaternion fields. The combination of thefield equations (23) together with the restrictions (7) on the spin-affine

connection then constitutes the quaternionic form for the differential relation

ships for the metric field of a Riemannian space.

While the explicit structure of TL in the field Eqs. (23) is an

important aim of the present programme of study, the main object of this

paper has been to show that the second-rank tensor form (2) of Einstein's

field equations does indeed decompose into two separate quaternion field

equations. These transform in co-ordinate space like vector field equations

while they behave like second-rank spinors of the type if] S ''j under trans-

formations in spinor space. The separate field Eqs. (23a) and (23b) are,

in fact, the time-reversals of each other. Thus, the "factorization" (23) of

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Einstein's Eqs, (2) is somewhat analogous to the factorization of the Klein-

Gordon equation

_ o . . fa*, 9 v? = -mX

into the two time-reversed two-component Dirac spinor equations in >\

and "V , The spinor form of Dirac1 s equation is, of course, more general

than the Klein-Gordon equation because 1) it entails extra degrees of

freedom in the spinor variables and 2) it allows non-reflection symmetric

interactions to appear - terms that would have no counterpart in a scalar

formalism. The quaternion forms (23) of the metrical field equations are

more general than the tensor form of Einstein's field equations for precisely

the same reasons. Since the postulate of relativity theory does not have to

do with reflection symmetry at all, it is contended that the quaternion form

(23) of the metrical field equations represents a full exploitation of the

principle of relativity in the construction of the field equations.that describe

gravitational forces.

4. A RECOVERY OF THE TENSOR FORM OF EINSTEIN'S

EQUATIONS

As a final step in this analysis it will be shown that a particular

combination of the time-reversed Eqs. (23a) and (23b) does indeed give back

the usual tensor form of Einstein's equations.

If we multiply Eq. (23a) on the right with^ and Eq. (23b) on the

left with Q.T and add the resulting equations, the following relation is

obtained:

\

(25)

qy (rTf)]y (rTf)

Thus, with the definitions (13) and (1) for the Ricci tensor and metric tensor,

-10-

Kq. (25) takes the Einstein form

where

is the "matter source" of the metrical field in the tensor form of these

equations.

5. CONCLUDING REMARKS

Summarizing, we have seen that the second-rank tensor form of

the Einstein field equations decomposes into a pair of quaternion equations

that are related to each other by time reversal. Thus the complexity of

the second-rank tensor field is reduced to a pair of vector fields while the

spinor degrees of freedom which were absent ("masked") in the single

tensor equations appear in the separated quaternion equations. The root

of this result lies, in fact, in the properties of the irreducible represent-

ations of the subgroup of rotations of the Einstein group. The lowest

dimensional .representations of this group in fact behave as second-rank

spinors of the quaternionic type ft JS Kl * . The latter represent a group

in which reflection symmetry elements are not defined while the larger

dimensional representations which underlie the Einstein1 s original tensor

formulation do admit reflection symmetry in space and time into the group.

Since the principle of relativity deals only with continuous trans-

formations in space and time, the quaternionic representations (which are

the lowest dimensional representations) then relate to the most general

expression of the theory. Indeed, within the formalism presented here,

we have seen that the usual form of Einstein's equations results only from

one particular way of combining the quaternion field Eq. (23a) with its time-

reversed Eq. (23b). It can then be conjectured that the separate use of

these equations should yield results that are out of the realm of prediction

of the tensor equations.

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Finally, it might be remarked that the quaternion form of the

metrical field equations lends itself in a natural way to a unification between

the inertial and gravitational manifestations of interacting matter. This is

because of the basic expression of the matter fields themselves in terms7

of the same spinor and quaternion variables. Studies of the possibility

of unification along these lines are currently in progress.

ACKNOWLEDGMENTS

I would like to thank Professor Abdus Salam and the IAEA for

hospitality at the International Centre for Theoretical Physics, Trieste,

during the summer, 1966, when this work was completed.

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REFERENCES AND FOOTNOTES

1. A. EINSTEIN, Math. Ann. 102, 685 (1930);

H. S. RUSE, Prac . Roy. Soc. (Edinburgh) j[7, 97(1937).

2. For an application of the quaternion formalism in general relativity toelectromagnetic theory, see

M. SACHS, Nuovo Cimento jU, 98 (1964). A recent survey of work

on the spinor formalism in a Riemannian space is given by CAP,

MAJEROTTO, RAAB and UNTEREGGER, Fortsch. d. Phys. l±, 205

(1966).

3. See, for example, ADLER, BAZIN and SCHIFFER, Introduction to

General Relativity (McGraw Hill Book Co., New York, 1965).

4. See E. SCHRODINGER, Space-Time-Matter (Cambridge University

Press, 1963) Chap. XII.

5. Note that the spin-affine connection-Q^ transforms covariantly as a

vector in co-ordinate space while it does not, by itself, transform

covariantly in spinor space. It is rather the combination ^'jfyu.+-£\/u?)rl

that behaves as a scalar in spinor space (as well as co-ordinate space).

This, of course, is analogous to the non-covariance of C , by itself

in co-ordinate space. Using the Palitini technique, the independent

variables .0 ,1^ in the variational procedure are continuous fields, but

they are not by themselves covariant entities. They are only used in

the mathematical procedure of the variational method to derive the

field equations.

6. Note that the field equations in the spin-affine connection will be some-

what altered from Eq. (24) when the generally covariant form of the

matter fields are included in the Lagrangian formalism. This is

because of the dependence of the latter differential forms on M u, .

The effect of this addition would then be to alter the relationship (7)

in such a way as to add the contribution from the spinor formalism of

the matter fields themselves.

7. M. SACHS, Nuovo Cimento 34, 81 (1964).

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